This week my PreAP Precal classes delved into polar coordinates and graphs. Last year, we did some converting back and forth between polar and rectangular equations and then spent a day on graphing polar equations using t-tables. While I did enjoy seeing where the petals and loops came from, we only got through 6 graphs and we didn’t really delve into recognizing the different equations and being able to do a quick sketch.

Then our school decided to teach Calculus BC this year and–oops!–turns out that skill is kind of an important thing. Plus I’m helping to write the PreAP curriculum guide so I wanted to make sure it was a good lesson. Well, at least a better lesson than what was offered (which was “graph these 5 equations at the same time on the TI 83, come up with generalizations like ‘if |a|<|b| then r can be negative which forms an inner loop’, then forget what you just generalized and use a t-table to graph these.”)

I knew I wanted to use desmos and I knew I wanted them to make generalizations. My coteacher and I decided that my first draft was asking a bit too much (like the original lesson did), then I found this great worksheet and decided to base my lesson on that, using Desmos instead of the graphing calculator because DUH.

Here is the word file and the desmos file. The students worked in pairs in the computer lab and except for mention to a few of them that you can use the sliders instead of typing in each equation (I left off “*like it says in the directions*” so I wouldn’t have to put a quarter in the sarcasm jar), I didn’t say anything to guide them. About 3/4 of them got through during the 50 minute class, with the rest having to finish up the last row for homework.

Now this class *hates* discovery learning with the force of a thousand suns. And they loved this! (or at least they told me they did and they were engaged and I didn’t hear any of the normal “I can’t learn unless you teach me.”) AND they made a lot of the discoveries I wanted them to, some of them more formalized than others, but all of them on the right path. Plus a few of them even had time at the end and chose to spend it playing with Desmos.

It also made the next day so much easier because they knew (a) what the graphs should look like (instead of my horrible attempts at drawing) and (b) what some parts of the equation did. I made this modest notetakermaker for the day:

It took most of the period to discuss and they had a matching worksheet for homework. Tomorrow we’ll practice graphing by hand and solving systems.

Now I’m sure you’re wondering how this lesson was bagel award-winning. As I may have previously mentioned, @bronilo16 is the world’s best math department chair who took on a fourth prep of PreAP Precal so I didn’t have 32+ students in 3 of my classes. I told her I would do all the prep work, but of course, since she’s awesome, we still spend a lot of time hashing out ideas (she’s also the Calculus teacher so she knows what to focus on and she has a ton of ways to make ideas easier–the notetakermaker was just a prettified version of her notes). So I’m running late to class Friday morning because of a meeting and she’s waiting in the hall with a bagel for me because of how awesome these lessons went!

That + a thank you letter in the mail from a current student + another student letting me know that I’m “quirky in a good way” because I “probably don’t hear that enough” = one awesome week of school!! Since I spent the first 6 weeks of school crying at least twice a week, this was quite the turnaround week! Maybe I will be able to make it until May!

But now the pressure is on and the next lesson is complex numbers….. Eek!